Twisted conformal algebra related to κ -Minkowski space
aa r X i v : . [ h e p - t h ] N ov Twisted conformal algebra related to κ -Minkowski space Stjepan Meljanac, ∗ Anna Pacho l, † and Danijel Pikuti´c ‡ Rudjer Boˇskovi´c Institute, Bijeniˇcka c.54, HR-10002 Zagreb, Croatia Dipartimento di Matematica ”Giuseppe Peano”,Universita degli Studi di Torino, Via Carlo Alberto, 10 - 10123 Torino, Italy
Twisted deformations of the conformal symmetry in the Hopf algebraic framework are constructed.The first one is obtained by Jordanian twist built up from dilatation and momenta generators. Thesecond is the light-like κ -deformation of the Poincar´e algebra extended to the conformal algebra,obtained by twist corresponding to the extended Jordanian r-matrix. The κ -Minkowski spacetimeis covariant quantum space under both of these deformations. The extension of the conformalalgebra by the noncommutative coordinates is presented in two cases, respectively. The differen-tial realizations for κ -Minkowski coordinates, as well as their left-right dual counterparts, are alsoincluded. PACS: 11.25.Hf, 16T05, 17B37, 17B81.
I. INTRODUCTION
The conformal symmetry is considered as the fundamental symmetry of spacetime. Even though it cannot describemassive particles and fields, many high-energy physics theories admit the conformal symmetry. It also includes twofundamental geometries - Poincar´e and de Sitter - as subcases. The conformal algebra c consists of the Lorentzgenerators M µν , translations P µ , dilatations D (which generate scaling transformations) and generators of the specialconformal transformations K µ . The metric tensor on the d -dimensional spacetime we denote as g µν (it does not needto be in the diagonal form, it only has to be symmetric and non-degenerate). The commutation relations of theconformal algebra c for d > d = 4 we deal with c = so (2 , M µν , M ρσ ] = i ( g µσ M νρ + g νρ M µσ − g µρ M νσ − g νσ M µρ ) , [ M µν , P ρ ] = i ( g νρ P µ − g µρ P ν ) , [ D, K µ ] = − iK µ , [ D, P µ ] = iP µ , [ K µ , P ν ] = 2 i ( g µν D − M µν ) , [ K µ , M νρ ] = i ( g µν K ρ − g µρ K ν ) , [ K µ , K ν ] = 0 , [ M µν , D ] = 0 , [ P µ , P ν ] = 0 (1)Together with the rise of the interest in the deformations of relativistic symmetries of spacetime [1, 2] the quantumdeformations of the conformal algebra have been investigated already in the 90’s [3]-[10]. After the introduction of the κ -deformed Poincar´e algebra (with M µν and P µ as its generators) with the dimensionfull deformation parameter κ , thesame classical r-matrix as in [1] ( r = iκ M ν ∧ P ν with special choice of the basis for which the metric tensor is g = 0)was used in the quantum deformation of Poincar´e-Weyl algebra [4]. Also, the deformations of the full D = 4 conformalsymmetries were introduced [5, 6], corresponding to the standard (i.e. time-like) version of the κ − deformation. Theso-called null-plane (light-cone) deformation of Poincar´e algebra [7] has been extended as well to the deformation ofPoincar´e-Weyl group [8] and to conformal group [9] as well. All of the above mentioned deformations of conformalsymmetry were corresponding to the κ -Minkowski spacetime noncommutativity: [ˆ x µ , ˆ x ν ] = i ( a µ ˆ x ν − a ν ˆ x µ ) in eithertime-like a µ a µ <
0, space-like a µ a µ > a µ a µ = 0 cases. The deformation parameter κ enters via a µ = κ τ µ , with τ ∈ {− , , } for the metric tensor with Lorentzian (mostly positive) signature.Since the Poincar´e-Weyl and conformal algebras contain dilatation D as additional generator, one can also consideranother classical r-matrix, r = iκ D ∧ P , as internal one for these algebras and the corresponding quantum deformations.For example in [10] the so-called Jordanian deformations of the conformal algebra (together with Anti de Sitter andde Sitter ones) were considered. One should also mention that the same (time-like) Jordanian r-matrix was also usedin the twisted deformation of the Poincar´e-Weyl algebra [11] as well as in the twisting of the inhomogeneous general ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] linear algebra [12] together with some applications found in field and gauge theories [13, 14]. A large class of Abeliantwists related to twisted statistics in κ -Minkowski spacetime was considered in [15].For the conformal algebra the twist type of the deformation was firstly considered in the Moyal-Weyl case [16]related with the Abelian classical r-matrix, r = iθ µν P µ ∧ P ν , corresponding to the constant-noncommutativity ofthe spacetime coordinates [ˆ x µ , ˆ x ν ] = iθ µν . The θ -deformation of the conformal algebra found some applications innoncommutative field theories, see e.g. [17] and has been extended also to deformation of the superconformal algebra[18] as well.Recently, the Jordanian deformations have gained in popularity in the applications in AdS/CFT correspondenceas some of the deformations of the Yang-Baxter sigma models were shown to preserve the integrability [19, 20].The Jordanian deformations related with the classical r-matrices (satisfying the classical or modified Yang-Baxterequation) were applied to the AdS part of the correspondence principle [21]. The κ -Minkowski spacetime was alsoconsidered in this context in [22].Our aim in this paper is to present the quantum deformations of the conformal algebra which are described by theclassical r-matrices satisfying classical Yang-Baxter equation (CYBE). For such cases the Drinfeld twists (satisfyingthe cocycle condition) provide explicitly the star-product in the algebra of spacetime coordinates. We are interestedin the Jordanian κ -deformation of the conformal symmetry and the light-like κ -deformation of the Poincar´e algebraextended to the conformal algebra which will correspond to Jordanian and extended Jordanian r-matrices, respectively.The deformed conformal algebra is considered as Hopf algebra with twisted coproducts and antipodes. Conformalinvariance is compatible with the κ -Minkowski spacetime which constitutes the covariant quantum space under bothof these deformations.The paper is organized as follows. We start, in Sec. II, with recalling the basics of the twist deformations and theconditions for noncommutative spacetime covariance with respect to the twisted symmetries. In Sec. III we considerthe Jordanian twist [11, 23] in the covariant form [24] providing, for the metric with Lorentzian signature, three kinds(time-, light- and space-like) of deformations of the conformal algebra depending on the type of the vector a µ . Thetwisted coalgebra sector is presented together with the corresponding κ -Minkowski spacetime realization consistentwith the Hopf-algebraic actions. Later, in Sec. IV, we investigate another twist [25–27] related by transposition tothe so-called extended Jordanian twist [28, 29]. The twist is built from only Poincar´e generators, therefore it providesthe extension of the light-like κ -deformation of the Poincar´e algebra to the conformal algebra. For the Poincar´esubalgebra, the a = 0 deformation reduces to the null-plane deformation of [7, 29]. The realization for κ -Minkowskicoordinates is also presented. In both cases, in Sec. III and IV, we include the cross-commutation relations betweenthe conformal algebra generators and the noncommutative coordinates. Also, the so-called left-right dual κ -Minkowskirealizations are constructed from the transposed twists in sections III and IV, respectively. The last section concludesthe paper with some remarks. II. TWIST DEFORMATIONS OF THE CONFORMAL ALGEBRA
The twist deformation framework of spacetime symmetries requires us to deal with the Hopf algebras instead ofLie algebras corresponding to the given symmetry. To introduce this notion, we need to extend the Lie algebra (weare interested in the conformal algebra c described by (1)) into the universal enveloping algebra U ( c ) which can beequipped with the Hopf algebra structures on its generators L = { M µν , P µ , D, K µ } in the following standard way:coproduct : ∆ ( L ) = L ⊗ ⊗ L (2)counit : ǫ ( L ) = 0 (3)antipode : S ( L ) = − L. (4)The above maps are then extended to the whole U ( c ). Such undeformed Hopf algebra can be seen as the conformalsymmetry of the usual Minkowski spacetime in the algebraic form given by an Abelian algebra of coordinate functions x µ ∈ A , which is itself a subalgebra of undeformed Heisenberg algebra H :[ x µ , x ν ] = 0 (5)[ x µ , P ν ] = iδ µν (6)[ P µ , P ν ] = 0 (7)The conformal algebra has the following representation: M µν = − x µ P ν + x ν P µ D = x · PK µ = 2 x µ ( x · P ) − x P µ = x µ D + x α M αµ (8)where P µ = g µν P ν . In general the compatibility of the spacetime with its symmetry in this “algebralized” setting isvia the action H ⊗ A → A of the Hopf algebra H on the spacetime (module) algebra A such that L ⊲ ( f · g ) = µ [∆( L )( ⊲ ⊗ ⊲ )( f ⊗ g )] . (9)The multiplication in the module algebra µ : A ⊗ A → A is compatible with the coproduct in the Hopf algebra∆ : H → H ⊗ H and L (1) = ǫ ( L ) ·
1, 1 ( f ) = f for L ∈ H and f ∈ A .One can easily check that the above condition (9) is satisfied for the undeformed spacetime described by Abelianalgebra (5) and the conformal Hopf algebra (1) with (2-4) as its symmetry and the condition reduces to the usualLeibniz rule: L ⊲ ( x µ · x ν ) = ( L ⊲ x µ ) x ν + x µ ( L ⊲ x ν ) , L = { M µν , P µ , D, K µ } (10)for any of the generators of the conformal algebra due to (2).For the deformation we will use the (Drinfeld) twist technique which will provide the deformation of the universalenveloping algebra of the conformal algebra U ( c ) as Hopf algebra H = ( U ( c ) , ∆ , ǫ, S ) . The twist F is, in general, aninvertible element of H ⊗ H satisfying cocycle and normalization conditions:(
F ⊗ ⊗ id ) F = (1 ⊗ F )( id ⊗ ∆) F , (11)( id ⊗ ǫ ) F = ( ǫ ⊗ id ) F = 1 ⊗ . (12)One gets the new Hopf algebra structure H F = (cid:0) U ( c ) , ∆ F , ǫ, S F (cid:1) via modifying the coproduct and antipode maps inthe following way: ∆ F ( L ) = F ∆( L ) F − , L ∈ H ǫ ( L ) = 0 , S F ( L ) = f α S (f α ) S ( L ) S (¯f β )¯f β . (13)Here we use the short notation for the twist as: F = f α ⊗ f α , F − = ¯f α ⊗ ¯f α . Both of the twisted deformationsconsidered in this paper will be compatible with the κ − Minkowski spacetime with the defining commutation relationsas: [ˆ x µ , ˆ x ν ] = i ( a µ ˆ x ν − a ν ˆ x µ ) (14)This algebra will constitute the module algebra over the deformed conformal Hopf algebra, i.e. it is its covariantquantum space.The cocycle condition (11) for the twist guarantees the co-associativity of the deformed coproduct ∆ F and alsoassociativity of the corresponding twisted star-product in the twisted module algebra A F ( A , µ ⋆ ): f ⋆ g = µ ⋆ ( f ⊗ g ) = µ ◦ F − ( ⊲ ⊗ ⊲ )( f ⊗ g ) = (¯f α ⊲ f ) · (¯f α ⊲ g ) (15)for f, g ∈ A . Additionally, to a given twist we can associate the so-called realization of noncommuting coordinatefunctions as follows: ˆ x µ = µ (cid:2) F − ( ⊲ ⊗
1) ( x µ ⊗ (cid:3) = (¯f α ⊲ x µ ) · ¯f α , x µ ∈ A (16)For the twisted case the compatibility between the deformed coproduct ∆ F and the ⋆ -product in the module algebrais analogous to (9): L ⊲ ( µ ⋆ ( f ⊗ g )) = µ ⋆ (cid:0) ∆ F ( L ) ( ⊲ ⊗ ⊲ ) ( f ⊗ g ) (cid:1) (17)In the literature this condition is known under twisted covariance and for example it was investigated in more detailin the context of the conformal algebra undergoing the Moyal-Weyl deformation with theta-deformed spacetime [16].The covariance under twisted symmetry was firstly proved in [2] for the Moyal-Weyl deformation of the Poincar´esymmetry and theta-spacetime. In the Hopf algebraic framework when the noncommutative spacetimes are Hopfmodules and their deformed symmetry is the Hopf algebra, the condition of covariance is automatically satisfied viathe requirements (9) and (17). It is also common to re-write the condition (9) as
L ⊲ ( f · g ) = ( L (1) ⊲ f ) · ( L (2) ⊲ g ) where Sweedler notation for the coproduct is used∆( L ) = L (1) ⊗ L (2) . III. JORDANIAN DEFORMATION OF THE CONFORMAL ALGEBRA
We can deform the conformal Hopf algebra U ( c ) (1, 2, 3, 4) with the Jordanian twist [11, 23, 24, 27]: F J = exp [ i ln(1 − a · P ) ⊗ D ] (18)where a · P = a µ P µ . The corresponding classical r-matrix is r = ia µ D ∧ P µ . For the metric with the Lorentziansignature we can distinguish here three cases when vector a µ can be either time-like, light-like or space-like, neverthelessthe formulae presented below are valid for arbitrary, symmetric and non-degenerate metric.For simplicity, we introduce the shortcut notation Z = 1 − a · P .The algebra relations (1) and counits (3) stay undeformed. The deformed coproducts are:∆ F J ( P µ ) = P µ ⊗ Z ⊗ P µ (19)∆ F J ( M µν ) = ∆( M µν ) − ( a µ P ν − a ν P µ ) Z − ⊗ D (20)∆ F J ( D ) = D ⊗ Z − ⊗ D (21)∆ F J ( K µ ) = K µ ⊗ Z − ⊗ K µ + 2[ a α M αµ + a µ D ] Z − ⊗ D − [2 a µ ( a · P ) + a P µ ] Z − ⊗ iD ( iD + 1) (22)And the deformed antipodes: S F J ( P µ ) = − Z − P µ (23) S F J ( M µν ) = − M µν − ( a µ P ν − a ν P µ ) D (24) S F J ( D ) = − ZD (25) S F J ( K µ ) = − Z { K µ − a α M αµ + a µ D ] D + [2 a µ ( a · P ) + a P µ ] iD ( iD + 1) } (26)The corresponding covariant quantum spacetime is the κ -Minkowski one (14) with the realization for coordinatesgiven via (16) as: ˆ x µJ = µ (cid:2) F − J ( ⊲ ⊗ x µ ⊗ (cid:3) = x µ − a µ D = x µ − a µ ( x · P ) (27)It is called right covariant realization [31, 32], and commutators with generators of the conformal algebra are:[ P µ , ˆ x νJ ] = − i ( g µν − a ν P µ )[ D, ˆ x µJ ] = − ix µ [ M µν , ˆ x λJ ] = i ( x µ g νλ − x ν g µλ )[ K µ , ˆ x νJ ] = i (2 x g µν − x µ x ν − a ν K µ ) (28)where x µ = ˆ x µJ + a µ D (29)Note that the commutators are closed in the conformal algebra and noncommutative coordinates ˆ x µJ .The spacetime algebra (14) obtained via (15) is invariant under the twisted conformal transformations which canbe seen from action of the conformal symmetry generators on the algebra of functions of κ -Minkowski coordinates,i.e. via the compatibility condition (17). One can check that indeed: L ⊲ (cid:2) µ ◦ F − J ( ⊲ ⊗ ⊲ )( x µ ⊗ x ν − x ν ⊗ x µ ) (cid:3) = L ⊲ [ i ( a µ x ν − a ν x µ )] (30)the twisted case of the Leibniz rule is satisfied for any of the generators L = { M µν , P µ , D, K µ } .Transposed twist ˜ F J = τ F J τ is obtained from F J by interchanging left and right side of tensor product (i.e. τ ( a ⊗ b ) = b ⊗ a ), and it is also a Drinfeld twist satisfying cocycle (11) and normalization condition (12). A set ofleft-right dual generators of κ -Minkowski space can be obtained from transposed twist:ˆ y µJ = µ h ˜ F J − ( ⊲ ⊗ x µ ⊗ i = x µ (1 − a · P ) (31)Generators ˆ y µJ satisfy κ -Minkowski algebra with a µ → − a µ :[ˆ y µJ , ˆ y νJ ] = − i ( a µ ˆ y νJ − a ν ˆ y µJ ) (32)and they commute with generators ˆ x µJ : [ˆ x µJ , ˆ y νJ ] = 0 (33) IV. LIGHT-LIKE κ − DEFORMATION OF POINCAR´E ALGEBRA EXTENDED TO THE CONFORMALALGEBRA
For the purpose of this section we consider the metric tensor with Lorentzian signature and use mostly positive signconvention, i.e. g µν = diag ( − , + , + , ..., +). We deform the conformal Hopf algebra U ( c ) (1, 2, 3, 4) with the twist F LL leading to light-like κ -deformation of Poincar´e algebra [14, 25, 27, 29] (which is related to extended Jordaniantwist [28],[30]) : F LL = exp (cid:20) − ia α P β ln(1 + a · P ) a · P ⊗ M αβ (cid:21) (34)Above twist satisfies the cocycle condition (11) [27] with the light-like vector a µ [25, 27] and the classical r-matrix is r = a µ M µν ∧ P ν . We also introduce the following notation:˜ Z = 11 + a · P , m µ = a α M αµ (35)Coproducts: ∆ F LL ( P µ ) = ∆ ( P µ ) + (cid:20) P µ a α − a µ (cid:18) P α + 12 a α P (cid:19) ˜ Z (cid:21) ⊗ P α (36)∆ F LL ( M µν ) = ∆ ( M µν ) + ( δ αµ a ν − δ αν a µ ) (cid:18) P β + 12 a β P (cid:19) ˜ Z ⊗ M αβ (37)∆ F LL ( m α ) = ∆ ( m α ) + a µ (cid:18) P α + 12 a α P (cid:19) ˜ Z ⊗ m α (38)∆ F LL ( D ) = ∆ ( D ) − P α ˜ Z ⊗ m α (39)∆ F LL ( K µ ) = K µ ⊗ (cid:20) δ αµ + P µ a α − a µ (cid:18) P α + 12 a α P (cid:19) ˜ Z (cid:21) ⊗ K α + nh (cid:16) a µ ( iD + ˜ Z ) − im µ (cid:17) P α + iM µα i − Dg µα o ⊗ m α + h iP µ g αβ − i ( δ αµ − a µ P α ˜ Z ) P β ˜ Z i ⊗ m α m β (40)Antipodes: S F LL ( P µ ) = − (cid:20) P µ + a µ (cid:18) P α + 12 a α P (cid:19) P α (cid:21) ˜ Z (41) S F LL ( M µν ) = − M µν − ( δ αµ a ν − δ αν a µ ) (cid:18) P β + 12 a β P (cid:19) M αβ (42) S F LL ( m µ ) = − m µ − a µ (cid:18) P α + 12 a α P (cid:19) m α (43) S F LL ( D ) = − ˜ Z − D (44) The explicit relation between the twist (34) and the standard extended Jordanian twist corresponding to light-like case (up to thetransposition) is presented in detail in sect. VIII B in ref. [27] S F LL ( K µ ) = − (cid:20) δ γµ + P γ a µ − a γ (cid:18) P µ + 12 a µ P (cid:19) ˜ Z (cid:21) ×× n K γ + h (cid:16) a γ ( iD + ˜ Z ) − im γ (cid:17) P α + iM γα i S ( m α ) − DS ( m γ )+ iP γ S ( m ) − i (cid:16) δ αγ − a γ P α ˜ Z (cid:17) P β ˜ ZS ( m α m β ) o (45)In this case, the realization (16) is given as:ˆ x µLL = µ (cid:2) F − LL ( ⊲ ⊗ x µ ⊗ (cid:3) = x µ + a α M αµ = x µ (1 + a · P ) − ( a · x ) P µ (46)It corresponds to the natural realization of κ -Minkowski space [31, 32]. Commutators with generators of the conformalalgebra are: [ P µ , ˆ x νLL ] = − i [ g µν (1 + a · P ) − a µ P ν ][ D, ˆ x µLL ] = − ix µ [ M µν , ˆ x λLL ] = i (ˆ x µLL g νλ − ˆ x νLL g µλ − a µ M νλ + a ν M µλ )[ K µ , ˆ x νLL ] = i (2 x g µν − x µ x ν + a µ K ν − g µν ( a · K )) (47)where x µ = ˆ x µLL − a α M αµ (48)Note that, like in Jordanian case, above commutators (47) are also closed in the conformal algebra and noncommutativecoordinates ˆ x µLL .Let us also comment on the fact that even though the above twist (34) is written in a covariant form (valid for the a µ as time-, light- and space-like vector) it satisfies the cocycle condition (11) only for the light-like case, a = 0 [25, 27].Therefore, only in this case it corresponds to an associative star-product (15) of κ -Minkowski coordinates (14). Thetwo remaining cases (time- and space-like) lead to a deformations of κ -Snyder type with non-associative star product.Again, one can easily check that the noncommutative spacetime (14) is invariant under this twisted conformalsymmetry via analogous condition as (30) in section III: L ⊲ (cid:2) µ ◦ F − LL ( ⊲ ⊗ ⊲ )( x µ ⊗ x ν − x ν ⊗ x µ ) (cid:3) = L ⊲ [ i ( a µ x ν − a ν x µ )] (49)Transposed twist ˜ F LL = τ F LL τ is obtained from F LL by interchanging left and right side of tensor product, andit is also a Drinfeld twist satisfying cocycle (11) and normalization condition (12). A set of left-right dual generatorsof κ -Minkowski space can be obtained from transposed twist:ˆ y µLL = µ h ˜ F − LL ( ⊲ ⊗ x µ ⊗ i = x µ + ( a · x ) P µ − a µ (cid:16) D + a · x P (cid:17) ˜ Z (50)Generators ˆ y µLL satisfy κ -Minkowski algebra with a µ → − a µ :[ˆ y µLL , ˆ y νLL ] = − i ( a µ ˆ y νLL − a ν ˆ y µLL ) (51)and they commute with generators ˆ x µLL : [ˆ x µLL , ˆ y νLL ] = 0 (52)Realizations ˆ y µJ and ˆ y µLL cannot be expressed in terms of x µ and generators of conformal algebra (whereas realizationsˆ x µJ and ˆ x µLL are expressed in terms of these generators).Note that ˆ x µJ (eq.(27)) and ˆ x µLL (eq.(46)) are different realizations of κ -Minkowski space ˆ x µJ = ˆ x µLL , related bysimilarity transformation. There is also another point of view, so that, for a = 0, ˆ x µJ and ˆ x µLL can be identified,but generators x µ and generators of conformal algebra have different realizations in two cases (sections III and IV),related by similarity transformation. In this case, let us denote as ( x µJ , P µJ ) and ( x µLL , P µLL ) two pairs of commutativecoordinates and momenta, each satisfying undeformed Heisenberg algebra (5)-(7), which are related by similaritytransformation: P µJ = (cid:18) P µLL + a µ P LL (cid:19) ˜ Z LL (53) P µLL = (cid:18) P µJ − a µ P J (cid:19) Z − J (54) x µJ = [ x µLL + a µ ( x LL · P LL )] ˜ Z − LL − ( a · x LL ) (cid:18) P µLL + a µ P LL (cid:19) (55) x µLL = [ x µJ − a µ ( x J · P J )] Z J + ( a · x J ) (cid:18) P µJ − a µ P J (cid:19) (56)where Z J ≡ − a · P J = 11 + a · P LL ≡ ˜ Z LL . (57)Hence, ˆ x µ = x µJ − a µ ( x J · P J ) = x µLL (1+ a · P LL ) − ( a · x LL ) P µLL and two sets of conformal generators, { P µJ , M µνJ , D J , K µJ } and { P µLL , M µνLL , D LL , K µLL } , are related by similarity transformation. V. CONCLUDING REMARKS
We have presented the two different κ -deformations of the conformal symmetry within the Drinfeld twist framework.Both twists provide the κ -Minkowski star product, therefore the κ -Minkowski spacetime stays covariant under thetwisted conformal symmetries. Thanks to the twist, we are also able to obtain the differential realization for thenoncommutative coordinates. The extension of the conformal algebra by the noncommutative coordinates is alsopresented and it includes the deformed phase space (deformed Heisenberg algebra) as subalgebra. For alternativepoint of view, where the phase space stays undeformed but the realizations of the conformal algebra generators aremodified, see e.g. [33]. Additionally, we have constructed, from transposed twists, another set of realizations satisfyingthe κ -Minkowski relations (with a µ → − a µ ). Both of the deformations presented in this paper (Jordanian and extendedJordanian) provide the so-called triangular deformation as the corresponding classical r-matrices satisfy the classicalYang-Baxter equation. Interestingly, the Jordanian and extended Jordanian deformations can be generated by other(than already mentioned) classical r-matrices. One can notice that the form of the conformal algebra (1) does notchange if we exchange the generators (see also similar comment in [6]) in the following way: P µ → K µ , K µ → P µ , D → − D, κ → κ (58)This allows us, to distinguish yet another classical r -matrix for the conformal algebra (besides for example the oneinvestigated in Sec. III for the Jordanian case r = ia µ D ∧ P µ ), i.e.: r = − i ˜ a µ D ∧ K µ with a new deformationparameter ˜ κ and ˜ a µ = ˜ κ a µ . Such r-matrix is satisfying the classical Yang-Baxter equation and the classical limit isobtained for ˜ κ → κ → ∞ ). The new Jordanian twist (18) with (58) for any ˜ a µ will satisfythe cocycle condition (11) as well. Formal expressions for the twisted deformation of coproducts and antipodes in (cid:0) U ( c ) , ∆ F , ǫ, S F (cid:1) (as twisted conformal Hopf algebra) generated by this r-matrix will stay the same up to (58).One way for interpreting the exchange in the deformation parameter κ → κ (related with a µ → ˜ a µ as above) couldbe the following. Instead of considering the minimal length, as it happens when introducing the noncommutativecoordinates ˆ x µ , we should consider the minimal momentum and introduce the noncommutative momenta ˆ p µ . Thisway the ˜ κ -deformation would appear in the momentum space [ˆ p µ , ˆ p ν ] = i (˜ a µ ˆ p ν − ˜ a ν ˆ p µ ) instead of (14). Other physicalconsequences of such exchange are still an open issue.Nevertheless, the deformations of the conformal symmetry introduced in this paper can be of interest in manyphysical applications. For example, the Jordanian deformations are also appearing in the context of AdS/CFTcorrespondence [21, 22], therefore the corresponding deformations of the conformal field theory part in the twistedframework could be of interest as well. Another point to consider would be, for example, the extension of thedeformations introduced in this paper to the supersymmetric case, as it was already considered for the Moyal-Weyldeformation of the conformal superalgebra [18]. Additionally extending the presented framework into the Hopfalgebroid language [24] would allow to introduce yet another example for the twisted deformation of Hopf algebroidsas well. Also, the deformations of the conformal symmetry presented here could be considered as a starting pointin the study on deformed (noncommutative) cosmology. Recently a short review on models of the inflating Universebased on conformal symmetry was presented [34]. The straightforward way to make them noncommutative wouldbe to introduce the star-product (15) related with the twists in the conformally invariant actions corresponding todifferent models. This way one could investigate, for example, if introducing the deformation parameter (as quantumgravity scale) would have any influence on the scale-invariance of the power spectrum of the scalar perturbations.Our results, however, provide only the starting point for such investigations. Acknowledgements
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